\( 2 x^{2}+x+3=0 \) The solution set is \( \} \). Use the quadratic formula to solve the equation. (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)

To solve the quadratic equation \( 2x^{2} + x + 3 = 0 \) using the quadratic formula, follow these steps: The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2 \), \( b = 1 \), and \( c = 3 \). 1. **Calculate the discriminant (\( \Delta \)):** \[ \Delta = b^2 - 4ac = 1^2 - 4 \cdot 2 \cdot 3 = 1 - 24 = -23 \] Since the discriminant is negative, the solutions will be complex numbers. 2. **Substitute the values into the formula:** \[ x = \frac{-1 \pm \sqrt{-23}}{2 \cdot 2} = \frac{-1 \pm i\sqrt{23}}{4} \] where \( i \) is the imaginary unit. 3. **Write the solution set:** \[ \left\{ \frac{-1 + i\sqrt{23}}{4}, \; \frac{-1 - i\sqrt{23}}{4} \right\} \] **Final Answer:** \[ \left\{\, \dfrac{-1 + \sqrt{-23}}{4},\; \dfrac{-1 - \sqrt{-23}}{4} \,\right\} \]